I'm sorry to report that Alan Baker died yesterday after a major stroke a few days ago. I knew him first when I was an undergraduate going to his number theory lectures, and later I was a colleague of his both in the Cambridge mathematics department and at Trinity College.

He became famous when he proved a far-reaching generalization of the Gelfond-Schneider theorem, which answered Hilbert's seventh problem. Hilbert's problem was the following. Suppose that a and b are two real numbers such that a is algebraic and not equal to 0 or 1, and b is irrational and also algebraic. Must a raised to the power b be transcendental (meaning that it can't be a root of a polynomial with integer coefficients)? For example, must 2 raised to the power the square root of 2 be transcendental? If I remember correctly, Hilbert believed that this was one of the hardest of his problems, but in fact it was one of the first to be solved.

One can reformulate the theorem by taking logs. It is equivalent to saying that if a_1 and a_2 are algebraic (and not 0 or 1) and b_1 is algebraic and irrational, then b_1log(a_1)-log(a_2) cannot be zero. We can express this more symmetrically by saying that if a_1 and a_2 are algebraic and b_1 and b_2 are algebraic with an irrational ratio b_1/b_2, then b_1log(a_1)+b_2log(a_2) cannot be zero.

Yet another way of expressing the result is to say that if a_1 and a_2 are algebraic numbers and their logarithms are linearly independent over the rationals, then these logarithms must in fact be linearly independent over the algebraic numbers.

Alan Baker extended this to an arbitrary number of algebraic numbers. That is, if a_1,...,a_k are algebraic and log(a_1),...,log(a_k) are linearly independent over the rationals (meaning that no non-trivial rational combination gives zero), then they are linearly independent over the algebraic numbers (meaning that no non-trivial linear combination with algebraic coefficients gives zero). This immediately implies the transcendence of all sorts of other numbers.

In fact, Baker did more: he gave a lower bound for how far away an algebraic combination of the logs of the a_i had to be from zero (in terms of various "heights", which tell you how complicated a polynomial you need to demonstrate that a number is algebraic). This had important applications to Diophantine equations.

One result I like of Baker's is an effective version of a consequence of Roth's theorem. I won't go into details, but a famous theorem of Roth implies that for every c>2 there is a constant delta>0 such that if p and q are positive integers and a is the cube root of 2, then |a - p/q| must be at least delta/q^c. Liouville's theorem, which is much easier, shows that it must be at least 1/q^3 or something similar. A problem with Roth's theorem is that it is ineffective: it does not tell you how small the constant delta needs to be. Baker managed to prove that if c=2.995 then you can take delta to be 0.000001. I like it because it is apparently such a negligible improvement over what you get from Liouville's theorem (indeed, q has to be very large before it gives any improvement at all), but that of course reflects how difficult the problem is.

There's a great story on this very topic that Szemerédi likes to tell about his youth. From his St Andrews biography:

At this stage Szemerédi did not have a doctorate and, given that Hungary now had strong links with Russia and Alexander Gelfond was a world leading number theorist, Szemerédi applied to do research in Moscow with Gelfond as his thesis advisor. However, an unfortunate spelling error (an "a" instead of an "o") meant that when he arrived in Moscow in 1967 he discovered that he had been assigned to be Israil Moiseevic Gelfand's student:

'I realized immediately that this was not for me, and Gelfand also realized it and advised me not to do mathematics anymore, telling me: "Just try to find another profession; there are plenty in the world where you may be successful." I was twenty-seven years old at the time, and he had all these star students aged around twenty, and twenty-seven was considered old!'

Gelfand's type of mathematics did not suit Szemerédi at all but he did not want to give up and return to Hungary as a failure. He did have a piece of luck, however, for around the end of his first year as a doctoral student in Russia, there was a conference in Debrecen in Hungary which Gelfond attended. Szemerédi, as a Hungarian student who could speak Russian, was assigned to look after him and show him round. Once Gelfond heard of the error that had occurred, he said that when he returned to Moscow he would arrange for Szemerédi to become his student. Sadly, this was not to be since Gelfond died two months after returning to Moscow. Szemerédi continued as Gelfand's student, but was allowed to write his thesis on combinatorics.